Question:
Suppose the fractal dimension $1\le d_c\le2$ of a planar parametric curve $c(t) := (x(t),y(t))$ is given;
can any nontrivial estimates for the fractal dimensions $d_x$ of $x(t)$ and $d_y$ of $y(t)$ be given?
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asked Feb 8, 2014 at 16:57
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1$\begingroup$ There is some interesting work computing the coordinate functionals for the standard space-filling curves $d_c=2$. One example: projecteuclid.org/euclid.rae/1285689164 $\endgroup$– Gerald EdgarCommented Mar 4, 2014 at 20:17
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1 Answer
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No nontrivial estimates, at least if the definitions are right.
Say let's define $$\ell_\alpha c=\sup\,\left\{\,\sum_{t_0<\dots<t_n}|c(t_i)-c(t_{i-1})|^\alpha\,\right\}$$ and $$\dim c=\sup\{\,\alpha\mid \ell_\alpha c<\infty\,\}.$$
Then from Pythagorean theorem we get $$\dim c=\max\{\dim x, \dim y\}.$$
answered Feb 8, 2014 at 18:09